Vol. 303, No. 2, 2019

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Polarization, sign sequences and isotropic vector systems

Gergely Ambrus and Sloan Nietert

Vol. 303 (2019), No. 2, 385–399
Abstract

We determine the order of magnitude of the $n$-th ${\ell }_{p}$-polarization constant of the unit sphere ${S}^{d-1}$ for every $n,d\ge 1$ and $p>0$. For $p=2$, we prove that extremizers are isotropic vector sets, whereas for $p=1$, we show that the polarization problem is equivalent to that of maximizing the norm of signed vector sums. Finally, for $d=2$, we discuss the optimality of equally spaced configurations on the unit circle.

Keywords
polarization problems, discrete potentials, Chebyshev constants, isotropic vectors sets, tight frames, vector sums
Primary: 52A40
Secondary: 41A17